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Theorem suctrALTcf 29014
Description: The sucessor of a transitive class is transitive. suctrALTcf 29014, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 29015, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALTcf  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALTcf
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4485 . . . . . . . 8  |-  A  C_  suc  A
2 id 19 . . . . . . . . 9  |-  ( Tr  A  ->  Tr  A
)
3 id 19 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
4 simpl 443 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
53, 4syl 15 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
6 id 19 . . . . . . . . 9  |-  ( y  e.  A  ->  y  e.  A )
7 trel 4136 . . . . . . . . . 10  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
873impib 1149 . . . . . . . . 9  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
92, 5, 6, 8syl3an 1224 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
10 ssel2 3188 . . . . . . . 8  |-  ( ( A  C_  suc  A  /\  z  e.  A )  ->  z  e.  suc  A
)
111, 9, 10eel0321old 28803 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
12113expia 1153 . . . . . 6  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
13 id 19 . . . . . . . . 9  |-  ( y  =  A  ->  y  =  A )
14 eleq2 2357 . . . . . . . . . 10  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1514biimpac 472 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
165, 13, 15syl2an 463 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
171, 16, 10eel021old 28778 . . . . . . 7  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1817ex 423 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
19 simpr 447 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
203, 19syl 15 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
21 elsuci 4474 . . . . . . 7  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
2220, 21syl 15 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
23 jao 498 . . . . . . 7  |-  ( ( y  e.  A  -> 
z  e.  suc  A
)  ->  ( (
y  =  A  -> 
z  e.  suc  A
)  ->  ( (
y  e.  A  \/  y  =  A )  ->  z  e.  suc  A
) ) )
24233imp 1145 . . . . . 6  |-  ( ( ( y  e.  A  ->  z  e.  suc  A
)  /\  ( y  =  A  ->  z  e. 
suc  A )  /\  ( y  e.  A  \/  y  =  A
) )  ->  z  e.  suc  A )
2512, 18, 22, 24eel2122old 28805 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2625ex 423 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2726alrimivv 1622 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
28 dftr2 4131 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2928biimpri 197 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
3027, 29syl 15 . 2  |-  ( Tr  A  ->  Tr  suc  A
)
3130iin1 28639 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    C_ wss 3165   Tr wtr 4129   suc csuc 4410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-uni 3844  df-tr 4130  df-suc 4414
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